Project 1 uses a developmental cognitive approach to understand core mathematical and cognitive processes related to mathematics disability (MD). Early cognitive developmental studies of MD often did not account for possible confounding effects of co-morbid learning disorders, such as reading disability (RD). More recent studies have begun to take other learning disorders into account, but have not investigated math disability in relation to behavioral disorders such as attention-deficit hyperactivity disorder (ADHD) that have substantial overlap with learning disorders. Very few studies are longitudinal, particularly in the early to mid-primary grades, which limits knowledge of factors related to the stability of patterns of mathematical deficits, relationships between acquisition of fact retrieval skills and later development of complex mathematical algorithms, and the rate of growth in different skill domains. Therefore, we propose to examine mathematical processes and their hypothesized related cognitive competencies in carefully ascertained community-based samples of typically developing children and children with MD, RD, and MD-RD, in which ADHD is also ascertained.. We will examine the speed and accuracy of fact retrieval and speed and accuracy of calculation, strategies used during calculation (e.g, counting, direct retrieval of facts from memory), approximate calculation or estimation of quantities, conceptual number knowledge, speed and accuracy of written multi-digit calculation, and solving complex word problems. We will examine the relationship between these domains of mathematical processing and supporting cognitive competencies, including working memory and executive control processes. Within a longitudinal design spanning third through fifth grades, we propose to identify possible empirically-based subtypes of MD, examine change in patterns of mathematical processes and supporting competencies in the MD subtypes, and ascertain the stability of MD subtypes. In relation to Project 2, we will examine predictors of response to mathematical interventions targeting fact retrieval, multi-digit calculation, estimation, and solving complex word problems. In relation to Projects 3 and 4, we will assess core mathematical processes and supporting cognitive competencies in relation to neuroimaging studies of fact retrieval, calculation, and estimation and will examine functional neuroimaging findings in empirically-derived subtypes of MD. Our approach is innovative in that it integrates careful case ascertainment and model-driven assessment of core mathematical skills and supporting cognitive competencies with studies of functional neuroimaging and intervention in children with MD.